Oh, I love this thread. Those diarrhea couplets are magnificent and bring back such fond playground memories.
I’m surprised nobody’s mentioned this classic, accompanied by strategic pointing/gestures at the appropriate body parts - I’m going to go ahead and assume you know which ones:
Milk
Milk
Lemonade
'Round the corner
Fudge is made!
Build a bonfire, build a bonfire,
Put the teachers on the top,
Put the headmaster in the middle,
Then burn the fuckin* lot!
(*actually, I think we were too chicken to sing this version, though we all did know that was what the word really should be, we actually tended to sing ‘bleedin’ or even ‘bleepin’.)
Also:
Charlie 'ad a pigeon, a pigeon, a pigeon,
Charlie 'ad a pigeon a pigeon 'e 'ad,
It flew in the day and it flew in the night,
And when it got back it was covered in-
Charlie 'ad a pigeon… repeat ad nauseam.
I remember a bunch of my classmates driving one of the teachers into a screaming fit when someone started that on a coach trip back from France to England…
Arrogant Worms did a much longer version of this called “The Assumption Song”.
Yankee Doodle went to town,
Riding on his mother.
Every time they hit a bump,
He had a baby brother.
I must have lived in more innocent times, because our version went:
Yankee Doodle went to town
Riding on a turtle
Turned the corner just in time
To see a lady’s girdle!
We also had a counting song:
10 little devils all dressed in red
Trying to get to Heaven on the end of a thread
The thread string broke and down they all fell
Instead of going to Heaven they all went to
9 Little devils all dressed in red…
You could keep that song going for quite a while depending on how many devils you started with. It was almost as bad as the 99 Bottles song.
A ten-year-old thread to bring out the ten-year-olds in all of us…
Salvation Army, Salvation Army
Put a nickel in the drug, save another drunken bum.
Salcvation Army, Salvation Army.
Put a nickel in the drum and you’ll be saved.
Bars close at 12:00 (boooooooo)
Bars reopen at 12:01 (yeaaaaaaaaah).
Repeat chorus
No drinks sold to children (Booooooooo)
Children under 2 (Yeaaaaaah).
Repeat chorus.
Not last night, but the night before
24 (insert favorite racial or homophobic slur)'s came knocking at my door
Went upstairs to get my gun
tripped over the pee pot on the run
I couldn’t swim, I couldn’t float
A big, bad turd went down my throat
Went downstairs to take a leak
tripped o’er the stove and burnt my dick
Went outside to cool it off
Those damn (insert favorite racial or homophobic slur)'s ate it off!
Cragmont Cola is the drink
to pour downt he kitchen sink
looks like vomit, tastes like ink
Cragmont Cola sure does stink!
I’ve been working on the railroad all my live long day
I’ve been working ont he railroad just to pass the time away
Can’t you hear the whistle blowing, rise up so early in the morn,
Can’t you hear the whistle blowing? Dinah blow my horn.
(2x)
Dinah woncha blow
Dinah woncha blow
Dinah woncha blow my horn
Coming in her mouth in the morning
Coming in her ass at ni-i-i-i-ght
Coming in her mouth in the morning
Coming in her ass at night
And she says
Fee-Fi-Fiddly-eye-oh
Fee-Fi-Fiddly-eye-oh-oh-oh
Fee-Fi-Fiddly-eye-ohhhhhhh
Shooting it down her throat!
Two Irishman, Two Irishman were digging in a ditch
One called the other a dirty son of a…
Peter Murphy had a dog, such a fine dog was he
Along came another and bit him on the…
Cocktails, gingerale, twenty-five cents a glass
And if you don’t like it you can shove it up your…
Ask me no questions, I won’t tell you any lies
And if you get hit with a bucket of shit you better close your eyes.
We had variants of some that have been posted:
Batman, took me to the movies
Batman, it was real groovy
Batman, took me to the park,
Batman, kissed me in the dark
Batman, put it in real easy
Batman, pulled it out real greasy …
Batman, Batman, Batman
Great big globs of greasy, grimey, gopher’s guts
Mutilated monkey meat
Little birdie’s twisted feet
All wrapped up in one little pack to eat
and I forgot my spoon…
but I have a straw!
You yellow-bellied sapsucker
mother-fucker cock sucker
two-balled bitch
Every time I see you
makes my two-balls itch
I can’t remember the whole thing but we had a Battle Hymn of the Republic parody:
My eyes have seen the glory of the burning of the school
We have tortured every teacher, we have broken every rule…
[sung to the tune of “That’s Amore” …]
When your balls hit the floor / Like a B-54
That’sa rupture …
When they fall out of bed / Like they’re made out of lead
It’s a rupture …
…
When you’re mowing the lawn / And you’re singing this song
And your balls hit the blade / And out comes lemonade
… Iiiiit’s a ruuupture …
Hallelujah, hallelujah
Throw a nickel on the drum, save another drunken bum.
Hallelujah, hallelujah
Throw a nickel on the drum, and you’ll be saved.
Hallelujah, hallelujah
Throw a nickel on the floor, save another dirty whore.
Hallelujah, hallelujah
Throw a nickel on the drum, and you’ll be saved.
Another one to the Tune of “Thats Amore”
When a thread says goodbye, and it just will not die, That’s a Zombie.
Twice it has been revived, but it just aint alive, That’s a Zombie.
We sang it like this when I was a kid:
My eyes have seen the glory of the burning of the school
We have tortured every teacher, we have broken every rule
We played poker, and we even played pool
In the cellar of the burning school.
Although, I must admit, I do like the idea of drowning the principal. 
We sang this in scout camp:
I wear my pink pajamas in the summer when it’s hot.
I wear my flannel nighties in the winter when it’s not.
And sometimes in the springtime, and sometimes in the fall
I’ll jump right in between the sheets with nothing on at all.
Glory, glory, hallelujah.
Glory, glory, what’s it to ya?
Balmy breezes blowin’ through ya,
With nothing on at all.
Sung to the tune of “I’m looking over a four leaf clover.”
I’m looking over my dead dog rover
That I over-ran with the lawn mower.
One leg is missing, the other’s all gone,
The third one’s spread all over the lawn,
But it really doesn’t matter anymore,
Cause the fourth one’s on the kitchen dooooooooor,
Oh I’m looking over, my dead dog rover
That I over-ran with lawn mower!
A pretty miss went out to P I I I I C K some flowers
She stepped in grass up to her A A A A N N KLES high
She saw a bird land on a T U U R R KEY feather
She wasn’t smart she let a F A A R R MER take her home.
Granny’s in the cellar,
Lordy how ya smell her,
She’s making pancakes on her greasy stove,
Her eyes are full of matter drippin in the batter,
With a great big (make a phlegm clearing sound) hanging from her nose.
I hate you, you hate me
I’m the one who killed Barney
With a shot tothe head from a loaded .44
No more Purple Dinosaur
Mine eyes have seen the glory of the coming of the lord
He is coming round the corner in a green-and-yellow Ford
He has one hand on the throttle and the other on a bottle
Of Pabst Blue Ribbon Beer!
Glory, Glory sock it to yah!
Teacher hit me with a ruler!
I bonked her on the bean with a rotten tangerine
And she kicked me out of school!
Love me tender, love me true; Elvis Presley is a ju!
(I didn’t know how to spell that last word or what it meant. I just repeated it from my older brother when he sang these lines. Our Dad heard me sing it and said, “Don’t say that–you’ll insult the Jews.”
These were verses I wrote myself:
*Last night as I lay on my pillow
I lay on my pillow last night
I reached in the dark for some bug spray
Next morning the bedroom was white!
Bring back, bring back, bring back my beige walls to me, to me…
Last night as I lay on my pillow
A gas main blew out under me
I heard a trerrific explosion
And landed outside in a tree!
Bring back, bring back, bring back my bedroom to me, to me…
Last night as I lay on my pillow
Last night when I lay all alone
I dreamt that I ate a marshmallow
Next morning my pillow was gone!
Bring back, bring back, bring back my pillow to me, to me…
Last night I programmed my computer
A thunderstorm started outside
A lightning bolt hit the main cable
Now all of my floppies are fried!
Bring back, bring back, bring back my software to me, to me…
Next door to me lives a young mother
Whose kids always get in her hair
Her four-year-old son found the scissors
And now her posterior’s bare!
Bring back, bring back, bring back my panties to me, to me…
My Bonnie lies over the ocean,
My Bonnie lies over the sea;
But while she lies over the ocean,
She isn’t here lying to me!*
The last verse of the “On Top of Old Smokey” one…
Early next morning
I heard a big blast
I knew it was Teacher
Going to heaven at last